Right Triangle
Trig
Section 4.3
Right Triangle Trig
• In the previous section, we worked with:
a)
b)
c)
d)
Angles in both radians & degrees
Coterminal angles
Complementary and supplementary angles
Linear and angular speed
• Now our focus is going to shift to triangles
Right Triangle Trig
Opp.
• In this section, we are going to be using only right
triangles
Ѳ
Adj.
Right Triangle Trig
• In this section, we are going to be using only right
triangles
Adj.
Ѳ
Opp.
Right Triangle Trig
• Using these three sides of the right triangle, we can
form six ratios that define the six trigonometric
functions.
Opposite
1. Sine θ =
Hypotenuse
Hyp.
4. Cosecant θ =
Opp.
Adjacent
2. Cosine θ =
Hypotenuse
Hyp.
5. Secant θ =
Adj.
Opposite
3. Tangent θ =
Adjacent
Adj.
6. Cotangent θ =
Opp.
Right Triangle Trig
• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent
Soh Cah Toa
Right Triangle Trig
3
• Find the value of the six trig functions of the
following triangle.
Ѳ
4
Right Triangle Trig
• Find the value of the six trig functions of the
following triangle.
Ѳ
15
Right Triangle Trig
• Find the value of the six trig functions of the
following triangle.
5
Ѳ
Right Triangle Trig
• You can also use a trig value to construct a right
triangle and find the values of the remaining trig
functions.
• E.g. Sin 𝜃
=
12
13
Right Triangle Trig
• Sketch a right triangle and find the values of the
remaining trig functions using the given information.
7
25
a) Cos 𝜃
=
b) Tan 𝜃
=3
c) Csc 𝜃
3
4
=
Right Triangle Trig
a) Cos 𝜃 =
7
25
Right Triangle Trig
b) Tan 𝜃
=3
Right Triangle Trig
c) Csc 𝜃 =
3
4
Right Triangle
Trig
Section 4.3
Right Triangle Trig
• Yesterday:
o Defined all six trig functions
o Found values of all six trig functions from right
triangles
o Constructed right triangles from a specific trig
value and found the remaining values
• Today
o Special right triangles
o Using a calculator
Right Triangle Trig
• Sketch a right triangle and find the values of the
remaining trig functions using the given information.
Cot 𝜃
=1
Right Triangle Trig
• Find the sine, cosine, and tangent of 45˚
Right Triangle Trig
• Using the equilateral triangle below, find the
sine, cosine, and tangent of both 30˚ and 60˚.
2
Right Triangle Trig
• These triangles are our two special triangles.
• We will use them throughout the year.
• The sooner you remember them, the easier your life
will be.
Right Triangle Trig
• From your triangles:
1
Sin 30˚ =
2
Csc 30˚ = 2
3
Cos 30˚ =
2
2 3
Sec 30˚ =
3
3
Tan 30˚ =
3
Cot 30˚ =
3
Right Triangle Trig
• From your triangles:
3
Sin 60˚ =
2
2 3
Csc 60˚ =
3
1
Cos 60˚ =
2
Sec 60˚ = 2
Tan 60˚ = 3
3
Cot 60˚ =
3
Right Triangle Trig
• From your triangles:
2
Sin 45˚ =
2
Csc 45˚ =
2
2
Cos 45˚ =
2
Sec 45˚ =
2
Tan 45˚ = 1
Cot 45˚ = 1
Right Triangle Trig
• The trig values of the angles 30˚, 45˚, and 60˚ are
values that we will be using from now until May.
• You will be expected to know these values from
memory.
• There will be quizzes that are non-calculator where
these values will be needed.
Right Triangle Trig
• Using a calculator
• On your calculators, you should see the three main
trig functions.
• Using these buttons, find the Sine 10˚
Right Triangle Trig
• Your calculator can also evaluate trig functions in
radians.
• To do this, you must switch the mode from degrees to
radians.
• Find the Cos
3𝜋
2
Right Triangle Trig
• Using your calculator, evaluate the following:
a) Tan 67˚
b) Sin 3.4
c) Sec 35˚
d) Cot
4𝜋
5
Right Triangle Trig
• In addition to radians and degrees, there is one more
type of unit we will be using throughout the year.
• Minutes and Seconds
o Most commonly used in longitude and latitude
o An angle in minutes and seconds would look like:
• 56˚ 8́ 10˝
Right Triangle Trig
• To convert minutes and seconds to degrees:
56˚ 8́ 10˝
Find the sum of the whole angle, the minutes divided by
60, and the seconds divided by 3,600
This will give you your angle in degrees
Right Triangle Trig
• Evaluate the following trig functions:
a) Sin 73˚ 56́
b) Tan 44˚ 28́ 16˝
c) Sec 4˚ 50́ 15˝
Right Triangle
Trig
Section 4.3
Right Triangle Trig
a) Find the remaining five trig functions if Tan Ѳ =
b) Find the exact value of the Cos 60˚ and Csc
c) Evaluate the Sec 37˚ to three decimal places
𝜋
3
5
12
Right Triangle Trig
• So far:
o Defined the six trig functions
o Created triangles from given trig values to find the remaining trig
values
o Used the special right triangles
• 30-60-90
45-45-90
o Used a calculator to evaluate trig functions of other angles
o Converted between degrees and minutes/seconds
• Today
o Find angles when given trig values
o Fundamental Identities
Right Triangle Trig
• So far, we have been using our trig functions to create
ratios.
• We can also use trig functions to solve for entire
triangles when given certain information.
• Must be given 1 side and one other part of the
triangle.
Right Triangle Trig
Solve for the remaining parts of the triangle.
8
35˚
Right Triangle Trig
Solve for the remaining parts of the triangle.
20˚
12
Right Triangle Trig
Solve for the remaining parts of the triangle.
5˚
18
Right Triangle Trig
• So far, the information we have been given has been
1 side and 1 angle.
• When we are given 2 sides, you must use your
calculator to evaluate the angle.
• This involves the inverse trig buttons on your calc.
o Sin−1
Cos−1
Tan−1
Right Triangle Trig
Solve for the remaining parts of the triangle.
10
7
Right Triangle Trig
Solve for the remaining parts of the triangle.
13
9
Right Triangle Trig
• Use the given information to solve for the remaining
parts of each triangle.
7
4
22˚ 11
Right Triangle Trig
7
4
Right Triangle Trig
22˚ 11
Right Triangle Trig
• Fundamental Trig Identities
o These are identities that we will use throughout the year
o It will be very beneficial to memorize them now as
opposed to struggling to remember them later
o We will go over 11 now, there will be over 30 throughout
the course of the year
o These are on page 283 of your book
Right Triangle Trig
• Reciprocal Identities
Sin θ
1
=
Csc θ
Csc θ =
1
Sin θ
Cos θ
1
=
Csc θ
Sec θ =
1
Sin θ
Tan θ
1
=
Csc θ
Cot θ =
1
Sin θ
Right Triangle Trig
• Quotient Identities
Tan θ
Sin θ
=
Cos θ
Cot θ
Cos θ
=
Sin θ
Right Triangle Trig
• Pythagorean Identities
Sin2 θ + Cos 2 θ = 1
1 + Tan2 θ = Sec 2 θ
2
2
1 + Cot θ = Csc θ
Right Triangle Trig
• We use the fundamental identities for 2 main
purposes:
1. To evaluate trig functions when given certain
information
2. Transformations (proofs)
Right Triangle Trig
• Let Ѳ be an acute angle such that Sin Ѳ = 0.6. Use the
fundamental identities to find the Cos Ѳ and Tan Ѳ.
Right Triangle Trig
• If Tan Ѳ = 5, find the remaining five trig functions of
Ѳ.
Right Triangle Trig
• If Csc Ѳ =
Ѳ.
13
,
2
find the remaining 5 trig functions of
Right Triangle Trig
• Transformations
o Similar to proofs from geometry
o Do not need to list reasons, just show steps
o Can only work with 1 side of the equation
o For this section, we will only be working with the
left side of the equation.
Right Triangle Trig
• Use the fundamental identities to transform the left
side of the equation into the right side.
Cos Ѳ Sec Ѳ = 1
Right Triangle Trig
• Use the fundamental identities to transform the left
side of the equation into the right side.
Cot Ѳ Sin Ѳ = Cos Ѳ
Right Triangle Trig
• Use the fundamental identities to transform the left
side of the equation into the right side.
(1 + Sin Ѳ) (1 – Sin Ѳ) = Cos2 Ѳ
Right Triangle Trig
• Use the fundamental identities to transform the left
side of the equation into the right side.
Tan θ+Cot θ
Tan θ
2
= Csc θ
Right Triangle Trig
• When dealing with word problems:
o Start by drawing a picture (preferably relating to
the problem)
o Label all sides of the triangle
o Determine what information you are looking for
o Set up a trigonometric ratio and solve
Right Triangle Trig
• Suppose you are standing parallel to the
Concord River. You turn and walk at a 60
degree angle until you reach the river,
which is a total distance of 130 ft. How
much farther down the river are you than
when you started?
Right Triangle Trig
• You are playing paintball with a friend when
you notice him sitting in a tree. The angle of
elevation to your friend is 23°. If the distance
between yourself and the tree is 60 ft, how
high up in the tree is your friend?
Right Triangle Trig
• Suppose you are putting in a basketball
hoop and need to know where to put the
“foul line”. You know that the height of the
hoop is 10 ft and the angle of elevation from
the “foul line” is 33.7°. Use this information to
determine where the foul line should go.
Right Triangle Trig
• You are a surveyor standing 115 ft from the
Washington Monument in Washington D.C.
You measure the angle of elevation to the
top of the monument to be 78.3°. How tall is
the Washington Monument?
Right Triangle Trig